# Is the sum of two (non-real) random variables necessarily a random variable?

Please note that I'm working with the following definition of random variable, which allows for a codomain other than $$\mathbb{R}$$.

Definition: Let $$(\Omega, \mathcal{F}, \mathbb{P})$$ be a probability space and $$(\Omega', \mathcal{F}')$$ a measurable space. Then a random variable is a $$\mathcal{F}/\mathcal{F}'$$-measurable mapping $$f: \Omega \to \Omega'$$.

I know that for $$(\Omega', \mathcal{F}') = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$$, the sum of two random variables on $$(\Omega, \mathcal{F}, \mathbb{P})$$ is also a random variable on the same probability space. Indeed, there are already proofs of this theorem here on Math.SE. But that leaves the following question open.

Question:

• Is it generally true that the sum of two random variables is a random variable?
• If not, what's a simple counterexample?
• Take a ball from an urn with red and green as possible outcomes and toss a coin (Heads or Tails). Then try to add the outcomes together – Henry May 21 at 16:06

Let $$\Omega'$$ be some set equipped with $$\sigma$$-algebra $$\{\varnothing,\Omega'\}$$ and let it be that there is no further structure present on $$\Omega'$$.
Then automatically every function $$f:\Omega\to\Omega'$$ is measurable but $$f+f$$ is not even defined.